Nonlinear perturbations of linear elliptic systems at resonance

We consider a semilinear system

$\Delta u$	$+ \lambda v+b_1(v)=f(x),\;\; x \in \Omega ,\quad \;\;\; u=0$$\mbox {\ \ \ \,\,for $x \in \partial \Omega $} ,$
$\Delta v$	$+\frac {\lambda ^2 _1}{\lambda } u+b_2(u) =g(x),\;\; x \in \Omega ,\quad v=0$$\mbox {\ \ \ \, for $x \in \partial \Omega $},$

whose linear part is at resonance. Here $\lambda >0$ and the functions $b_1(t)$ and $b_2(t)$ are bounded and continuous. Assuming that $tb_i(t)>0$ for all $t \in R$, $i=1,2$, and that the first harmonics of $f(x)$ and $g(x)$ lie on a certain straight line, we prove the existence of solutions. This extends a similar result for one equation, due to D.G. de Figueiredo and W.-M. Ni.